The outermost gravitationally bound orbit around a mass clump in an expanding Universe: implication on rotation curves and dark matter halo sizes

Abstract

Conventional treatment of cold dark matter halos employs the Navarro-Frenk-White (NFW) profile with a maximum radius set at $r=r_{200}$, where the enclosed matter has an overdensity of 200 times the critical density. The choice of $r=r_{200}$ is somewhat arbitrary. It is not the collapsed (virial) radius, but does give $r \sim$ 1 Mpc for rich clusters, which is a typical X-ray size. Weak lensing measurements, however, reveal halo radii well in excess of $r_{200}$. Is there a surface that places an absolute limit on the extension of a halo? To answer the question, we derived analytically the solution for circular orbits around a mass concentration in an expanding flat Universe, to show that an outermost orbit exists at $v/r = H$, where $v$ is the orbital speed and $H$ is the Hubble constant. The solution, parametrized as $r_2$, is independent of model assumptions on structure formation, and {\it is the radius at which the furthest particle can be regarded as part of the bound system}. We present observational evidence in support of dark matter halos reaching at least as far out as $r=r_2$. An interesting consequence that emerges concerns the behavior of rotation curves. Near $r=r_2$ velocities will be biased low. As a result, the mass of many galaxy groups may have been underestimated. At $r=r_2$ there is an abrupt cutoff in the curve, irrespective of the halo profile. An important cosmological test can therefore be performed if velocity disperion data are available out to 10 Mpc radii for nearby clusters (less at higher redshifts). For Virgo it appears that there is no such cutoff.